Universiteit Utrecht

Department of Mathematics

A unified approach to heavy traffic for asymptotically stable random walks

Vsevolod Shneer

22 April 2009

For families of random walks {Sk(a)} with ESk(a) = -ka < 0 we consider their maxima M(a) = supk ≥ 0 Sk(a). We investigate the asymptotic behaviour of M(a) as a → 0 for asymptotically stable random walks. This problem appeared first in the 1960's in the analysis of a single-server queue when the traffic load tends to 1 and since then is referred to as the heavy-traffic approximation problem. Kingman and Prokhorov suggested two different approaches which were later followed by many authors. We give two elementary proofs of the main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance. Such a generalisation will also be discussed during the talk. This is a joint work with Vitali Wachtel (University of Munich).

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Alexandra Babenko